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Maths-

General

Easy

Question

A piece of paper is in the shape of a square of side 1 m long. It is cut at the four corners to make a regular polygon of eight sides (octagon). The area of the polygon is

$2 (\sqrt{2} - 1) m^{2}$
$(\sqrt{2} - 1) m^{2}$
$\frac{1}{\sqrt{2}} m^{2}$
None of these

The correct answer is: $2 open parentheses square root of 2 minus 1 close parentheses m to the power of 2 end exponent$

Clearly, $blank x plus square root of 2 x plus x equals 1.$ So $comma blank x equals fraction numerator 1 over denominator 2 plus square root of 2 end fraction$
The required area $equals open parentheses 1 to the power of 2 end exponent minus 4 cross times fraction numerator 1 over denominator 2 end fraction x to the power of 2 end exponent close parentheses m to the power of 2 end exponent$
$equals open curly brackets 1 minus 2 fraction numerator 1 over denominator open parentheses 2 plus square root of 2 close parentheses to the power of 2 end exponent end fraction close curly brackets m to the power of 2 end exponent$
$equals open curly brackets 1 minus fraction numerator 1 over denominator open parentheses square root of 2 plus 1 close parentheses to the power of 2 end exponent end fraction close curly brackets m to the power of 2 end exponent$
$equals open square brackets 1 minus open parentheses square root of 2 minus 1 close parentheses to the power of 2 end exponent close square brackets m to the power of 2 end exponent equals 2 open parentheses square root of 2 minus 1 close parentheses m to the power of 2 end exponent$

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